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midterm_2_sample_solutions

# midterm_2_sample_solutions - MA 35100 SAMPLE MIDTERM EXAM#2...

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MA 35100 SAMPLE MIDTERM EXAM #2 SOLUTIONS Problem 1. Consider the matrix A = 1 1 1 1 2 5 1 3 9 . a. Compute its reduced row-echelon form. Show all of your work. b. Find a basis for ker( A ). What is the nullity of A ? c. Find a basis for im( A ). What is the rank of A ? Solution: (a) We compute the reduced row-echelon form for A . Subtract the first row from the second and third rows: 1 1 1 0 1 4 0 2 8 . Now subtract the second row from the first row, and twice the second row from the third row: rref( A ) = 1 0 - 3 0 1 4 0 0 0 . (b) The reduced row-echelon form above is the coefficient matrix for the system x 1 - 3 x 3 = 0 x 2 + 4 x 3 = 0 The kernel of A consists of solutions of this linear system. We see that x 3 = t is a free variable, so the general solution is in the form x 1 x 2 x 3 = t 3 - 4 1 i.e., 3 - 4 1 is a basis for ker( A ). The nullity of A is the dimension of the kernel of A , so null( A ) = 1.

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midterm_2_sample_solutions - MA 35100 SAMPLE MIDTERM EXAM#2...

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